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A few remarks on Pimsner–Popa bases and regular subfactors of depth 2

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  01 December 2021

Keshab Chandra Bakshi  and
Ved Prakash Gupta
Keshab Chandra Bakshi
Affiliation:
Chennai Mathematical Institute, Chennai, India. e-mail: [email protected]
Ved Prakash Gupta
Affiliation:
School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India
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Abstract

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

Keywords

Pimsner-Popa basesweak Hopf algebraregular subfactorunitary basisJones index

MSC classification

Primary: 46L37: Subfactors and their classification
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 586 - 602
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

In memory of Vaughan Jones, a true pioneer!

References

Bakshi, K. C., On Pimsner-Popa bases, Proc. Indian Acad. Sci. (Math. Sci.) 127(1) (2017), 117–132.CrossRefGoogle Scholar
Bakshi, K. C. and Gupta, V. P., On orthogonal systems, two sided bases and regular subfactors, New York J. Math. 26 (2020), 817–835.Google Scholar
Bakshi, K. C. and Kodiyalam, V., Commuting squares and planar subalgebras, J. Operator Theory 86(1) (2021), 145–161.Google Scholar
Bisch, D., Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, in Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Institute Communications, vol. 13 (American Mathematical Society, Providence, RI, 1997), 13–63.Google Scholar
Böhm, G., Nill, F. and Szlachányi, K., Weak Hopf algebras I. Integral theory and $C^*$ -structure, J. Algebra 221 (1999), 385–438.CrossRefGoogle Scholar
Ceccherini-Silberstein, T., On subfactors with unitary orthonormal bases, J. Math. Sci. 137(5) (2006), 5137–5160.CrossRefGoogle Scholar
David, M.-C., Paragroupe d’Adrian Ocneanu et algebre de Kac, Pac. J. Math. 172 (1996), 331–363.CrossRefGoogle Scholar
Goodman, F., de la Harpe, P. and Jones, V. F. R., Coxeter graphs and towers of algebras, MSRI Publ., vol. 14 (Springer, New York, 1989).CrossRefGoogle Scholar
Hong, J. H., A characterization of crossed products without cohomology, J. Korean Math. Soc. 32(2) (1995), 183–193.Google Scholar
Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 1–25.CrossRefGoogle Scholar
Jones, V. F. R. and Sunder, V. S., Introduction to subfactors , LMS LNS, vol. 234 (Cambridge University Press, Cambridge, 1997).Google Scholar
Longo, R., A duality for Hopf algebras and subfactors I, Comm. Math. Phys. 159 (1994), 133–150.CrossRefGoogle Scholar
Nikshych, D. and Vainerman, L., A characterization of depth 2 subfactors of $II_1$ factors, J. Funct. Anal. 171 (2000), 278–307.CrossRefGoogle Scholar
Nikshych, D. and Vainerman, L., Finite quantum groupoids and their applications, in New Directions in Hopf algebras, MSRI Publications, Mathematical Sciences Research Institute Publications, vol. 43 (2002), 211–262.Google Scholar
Nill, F., Szlachanyi, K. and Wiesbrock, H., Weak Hopf algebras and reducible Jones inclusions of depth 2: From crossed products to Jones towers, preprint math.QA/9806130 (1998).Google Scholar
Ocneanu, A., Quantized groups, string algebras and Galois theory for algebras, in Operator algebras and applications, vol. 2 (Warwick 1987) (Cambridge University Press, 1988), 119–172.CrossRefGoogle Scholar
Pimsner, M. and Popa, S., Entropy and index for subfactor, Ann. Sci. Ecole Norm. Sup. 19(3) (1986), 57–106.CrossRefGoogle Scholar
Popa, S., Orthogonal pairs of $\ast$ -subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), 253–268.Google Scholar
Popa, S., Notes on Cartan subalgebras of $II_1$ -factors, Math. Scand. 57 (1985), 171–188.CrossRefGoogle Scholar
Popa, S., Classification of amenable subfactors of type II, Acta Math. 172 (1994), 163–255.CrossRefGoogle Scholar
Popa, S., Asymptotic orthogonalization of subalgebras in $II_1$ -factors, Publ. RIMS Kyoto Univ. 55 (2019), 795–809.CrossRefGoogle Scholar
Popa, S., Shlyakhtenko, D. and Vaes, S., Classification of regular subalgebras of the hyperfinite $II_1$ -factor, J. Math. Pures Appl. 140 (2020), 280–308.CrossRefGoogle Scholar
Szymański, W., Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120(2) (1994), 519–528.CrossRefGoogle Scholar
Teruya, T. and Watatani, Y., Lattices of intermediate subfactors for type III factors, Arch. Math. 68(6) (1997), 454–463.CrossRefGoogle Scholar
Watatani, Y., Index for $C^*$ -subalgebras, Mem. Amer. Math. Soc. 83(424) (1990).Google Scholar